Moreover, solutions turn out to inherit polynomial mass-estimates from the initial condition, see e.g. Theorem 3.1 of [1]} for the case in \(\mathbb {R}^d\) . While throughout we do not assume more from the initial condition than having finite entropy with respect to \(\Pi _n\)  (which does imply the finiteness of the first moment) and unfortunately arbitrary curves \((\mathsf {P},\mathsf {J}^+,\mathsf {J}^-)\)  with finite \(\mathcal {I}_n\)  do not preserve moment estimates, we will include the statement for completeness.
